Probabilistic Genotype-Phenotype Maps Reveal Mutational Robustness of RNA Folding, Spin Glasses, and Quantum Circuits

Recent studies of genotype-phenotype (GP) maps have reported universally enhanced phenotypic robustness to genotype mutations, a feature essential to evolution. Virtually all of these studies make a simplifying assumption that each genotype—represented as a sequence—maps deterministically to a single phenotype, such as a discrete structure. Here, we introduce probabilistic genotype-phenotype (PrGP) maps, where each genotype maps to a vector of phenotype probabilities, as a more realistic and universal language for investigating robustness in a variety of physical, biological, and computational systems. We study three model systems to show that PrGP maps offer a generalized framework which can handle uncertainty emerging from various physical sources: (1) thermal fluctuation in RNA folding, (2) external field disorder in spin glass ground state finding, and (3) superposition and entanglement in quantum circuits, which are realized experimentally on IBM quantum computers. In all three cases, we observe a novel biphasic robustness scaling which is enhanced relative to random expectation for more frequent phenotypes and approaches random expectation for less frequent phenotypes. We derive an analytical theory for the behavior of PrGP robustness, and we demonstrate that the theory is highly predictive of empirical robustness.

Introduction.-Systems which take a sequence as input and nontrivially produce a structure, function, or behavior as output are ubiquitous throughout the sciences and engineering. In biological systems such as RNA folding [1][2][3][4][5][6][7][8][9][10][11], lattice protein folding [4], protein self-assembly [12,13], and gene regulatory networks [14,15], the relationship between genotype (stored biological information) and phenotype (observable or functional properties) can be structured as genotypephenotype (GP) maps, which have a rich history of computational and analytical investigation . Systems from physics and computer science have also been analyzed as GP maps, including the spin glass ground state problem [30], linear genetic programming [26], and digital circuits [31].
Despite being completely disparate systems, all of the GP maps above share a number of common structural features, most notably an enhanced robustness of the phenotypes to genotype mutations. Phenotypic robustness ρ n of a phenotype n is the average probability that a single character mutation of a genotype g which maps to n does not change the resultant phenotype n, averaged over all genotypes g mapping to n. Random assignment of genotype to phenotype predicts that ρ n ≈ f n [4], where f n is the fraction of genotypes that map to phenotype n. However, the systems mentioned above display substantially enhanced robustness, exhibiting the relationship ρ n ≈ a + b log f n f n with system-dependent constants a and b. It has been shown that, in evolution, this enhanced robustness facilitates discovery of new phenotypes [11,19,20,33] and is crucial for navigating fitness landscapes [5]. As a result, it is important to accurately quantify robustness and its relationship with phenotype frequency.
All of the GP map studies referenced above make the assumption that a genotype maps deterministically to a single phenotype. However, we argue that for most of the above systems, this is a major simplification. For instance, within a bulk sample of ∼ N mammalian cells, we expect to find ∼ N copies and ∼ N * 10 4 copies of a protein [34]. In vitro, such molecules often misfold [35], which is why cellular machinery exists to assist this folding and to degrade misfolded structures in vivo. By mapping a genotype to only the ground state energy structure, previous studies [1][2][3][4][5][6][7][8][9][10][11] make an implicit zero temperature approximation for the ensemble of molecules, even if the Gibbs free energy of an individual molecule itself is calculated within the folding software at finite temperature. Similarly, in studies of gene regulatory networks, spin glasses, linear genetic programs, and digital circuits, the systems investigated are small and do not interact with external networks or variables. These investigations assume that the environmental effect on the GP mapping of the subsystem of interest is static.
In this Letter, we introduce probabilistic genotypephenotype (PrGP) maps, in contrast to the above systems which we call deterministic genotype-phenotype (DGP) maps, which emerge as a limiting case of PrGP maps. The definitions of phenotypic robustness and transition probabilities retain the same physical meaning in PrGP maps as in DGP maps, and we emphasize that PrGP maps can handle disorder and uncertainty emerging from a variety of sources. To address the implicit zero temperature approximation in sequence-tostructure mappings (RNA, lattice protein folding, protein self-assembly), we study the folding of RNA primary sequences to a canonical ensemble of secondary structures corresponding to low-lying local free energy minima. To address external variable disorder with a known distribution, we study the zero temperature mapping of a spin glass bond configuration to its ground state with quenched external field disorder, building a phenotype probability vector using many replicas of the disordered field. This has implications for viral fitness landscape in- ference [36][37][38][39][40], where external fields, in part, model host immune pressure [39]. Lastly, to investigate inherent uncertainty in phenotypes, we introduce quantum circuit GP maps where uncertainty emerges from superposition and entanglement of classically measurable basis states. Our experimental realization of these quantum circuits on a 7-qubit IBM quantum computer also introduces measurement noise, which has a clear and unique effect on robustness. The PrGP map properties of the three model systems are summarized visually in Figure 1.
We observe that PrGP maps exhibit a novel biphasic scaling of robustness versus phenotype frequency which, for higher frequency phenotypes, resembles the ρ n ∝ log f n seen in DGP maps but is suppressed, and, for lower frequency phenotypes, settles closer to a linear relationship between ρ n and f n , suggesting that the lowest frequency phenotypes either appear sporadically throughout the GP map or are uniformly scattered at low probabilities throughout the genotype domain.
Theory.-Let Ω(g) = n represent the mapping of genotype g to phenotype n, where g is an element of S ,k , the set of all k sequences of length drawn from an alphabet of k characaters. A generalization of robustness is the transition probability φ mn , the average probability that a single character mutation of a genotype mapping to phenotype n will change the phenotype to m, with the average taken over all genotypes mapping to n. For DGP maps, φ mn is given by where I[·] is the indicator function, and nn(g) is the single character mutational neighborhood of sequence g. For PrGP maps, we weaken the indicator I[Ω(g) = n] to a probability p n (g) ≡ P[Ω(g) = n], which allows us to write where p(g) = (p 0 (g), p 1 (g), . . .) is the phenotype probability vector to which genotype g maps, and ∆ ,k is the set of all k (k − 1)/2 unordered pairs of sequences in S ,k which differ by exactly one character. The phenotype probability vector obeys the normalization conditions k f = g∈S ,k p(g) and 1 = n∈{phenotypes} p n (g) for all g ∈ S ,k , and phenotype robustnesses are given by the diagonal of the transition probability matrix, ρ n = φ nn . The phenotype entropy S(g) = − n∈{phenotypes} p n (g) log p n (g) of a genotype g is also useful for quantifying how deterministic or probabilistic a PrGP map is.
In DGP maps, a random null model [4] for robustness can be built by randomly assigning genotype-phenotype pairings while keeping the frequencies f constant. As a result, the probability of a single character mutation leading to a change from phenotype n to phenotype m is approximately φ mn ≈ f m for all m. For PrGP maps, a naive expectation can be built by letting all phenotype probability vectors equal the frequency vector, p(g) = f for all genotypes g. From eq. (2), one finds that φ mn = f m ; thus, the two random expectations are the same, even though they physically represent different scenarios.
RNA Secondary Structure Maps.-In RNA folding DGP map studies [1][2][3][4][5][6][7][8][9][10][11], the global free energy minimum secondary structure (reported as a "dot-bracket" string indicating polymer connectivity) was calculated for every RNA sequence of fixed length drawn from the alphabet of the four canonical nucleotides {A, C, G, U} (alphabet size k = 4). Here, we are interested in not only the global free energy minimum structures but also the lowlying local minima, and we additionally investigate the temperature-dependent behavior of the robustness. We use the RNAsubopt program from the ViennaRNA pack-  [41] to calculate the secondary structures and associated Gibbs free energies for the local free energy minima within 6 kcal/mol of the global free energy minimum (or all the nonpositive free energy local minima, if the global minimum is greater than −6 kcal/mol). Because of the increased computational time required to discover all the local minima within an energy range, we use a reduced alphabet of {C, G} for our main simulations with sequence length = 20. A validation study with = 12 and the full k = 4 alphabet is reported in the Supplemental Material [42]. Simulations for the = 20, k = 2 trials were conducted at 20 • C, 37 • C (human body temperature), and 70 • C. We take the low-lying local free energy minima structures to comprise a canonical ensemble at the simulation temperature, so the probability of RNA sequence g mapping to secondary structure n is determined from p n (g) = e −∆Gn/(RT ) /Z, where Z normalizes the vector. We then calculate the robustness, transition probabilities, and phenotype entropy distributions as detailed in the previous section. The DGP map limits of the PrGP map are also plotted for each temperature.
In Figure 2(a-b), we plot the relationship between robustness and frequency for the = 20, k = 2 RNA PrGP map and for the DGP map limiting cases for each simulation temperature (see Supplemental Material [42] for Perason and Spearman correlations). The DGP maps confirm the results of refs. [3,4], which emphasize that ρ n ∝ log f n for most phenotypes with significant elevation above the random null model [4] expectation. We find that there is little temperature dependence in DGP robustness calculations (see Supplemental Material [42]), suggesting that the effect of temperature does little to alter the exact ground state phenotype. However, our PrGP map results showcase a different robustness behavior. As the simulation temperature increases, there is a gradual but clear suppression of the robustness versus frequency relationship, as is apparent in both panels (a) and (b). We suggest this occurs due to two factors: firstly, though the ground state structure itself does not change much with temperature, the ground state becomes less stable relative to low-lying local minima, thereby increasing phenotype entropy, as evidenced by the entropy plots in the Supplemental Material [42]. As a result, for the corresponding p(g) ⊗ p(h) terms contributing to φ mn , probability mass is drawn away from the diagonals toward the off-diagonal transition probabilities. Secondly, as temperature increases, many low frequency (higher ∆G) phenotypes are discovered, increasing the number of phenotypes and drawing probability mass away from the more robust phenotypes.
For high frequency phenotypes, the PrGP map robustness is suppressed relative to the DGP map robustness, but is nonetheless substantially elevated above the random null expectation like in the DGP maps. However, for lower frequencies, the robustness behaves more like the random model; in the Supplemental Material, we see from a log-log plot of ρ n versus f n that robustness travels nearly parallel to the random null expectation, suggesting linear ρ n ∝ f n behavior up to a constant multiplicative factor. This biphasic robustness behavior becomes even clearer in the spin glass and quantum circuit PrGP maps. Off-diagonal transition probabilities maintained an approximate relationship φ mn ∝ f m for m = n, in concordance with DGP maps (see Supplemental Material [42]).
Spin Glass Ground State Maps.-In a previous spin glass [43,44] DGP map study [30], a zero temperature ±J spin glass on a random graph G(V, E) with Hamiltonian H(s; J) = − {i,j}∈E J ij s i s j − i∈V h i s i was considered. The genotype is the bond configuration where each J ij ∈ {−1, +1}, and the phenotype is the ground state configuration where each s i ∈ {−1, +1}. Degeneracies of the ground state were broken by the uniformly drawn, i.i.d. random external fields h i ∈ [−10 −4 , 10 −4 ] which were fixed for each simulation. In our spin glass PrGP map, we use a similar setup, but we are interested in the effect of external field disorder on robustness. We therefore incorporate the effects of Gaussian-distributed external fields h i ∼ N (h 0,i , σ 2 h ), where the uniformly distributed means h 0,i ∈ [−0.1, 0.1] are fixed across all realizations of the disorder for each simulation. To obtain accurate robustness measurements, we exactly calculate every ground state for spin glasses with |V | = 9, and |E| = 15 by exhaustive enumeration. We examine the effect of external field disorder by simulating 450 replicas of {h i } with variances σ 2 h = 0.001, 0.01, and 0.1 and fixed means {h 0,i }. Phenotype probability vectors for each genotype g ≡ J were constructed by tallying and normalizing the number of appearances of each ground state across each replica. Graph topology G(V, E) corresponding to data presented here, as well as validation trial data, are in the Supplemental Material [42].
In Figure 2(c-d), we plot robustness versus frequency of each ground state for each external field variance σ 2 h as well as the DGP map limiting case, which qualitatively reproduce the results of the earlier work [30] (see Supplemental Material [42] for Pearson and Spearman correlations). Trends similar to the RNA PrGP map are observed. Namely, as the disorder parameter (temperature for RNA and field variance for spin glasses) increases the uncertainty in the genotype-phenotype pairing, the phenotype entropy distribution shifts rightward (see Supplemental Material [42]), and the robustness versus frequency relationship becomes suppressed relative to the DGP map limit. Here, the spin glass results are more clearly suggestive of the proposed biphasic robustness relationship, especially apparent in panel (d). For the highest frequencies, the ρ n is substantially enhanced above the random null expectation and behavior close to the deterministic limit is observed. However, for the smallest frequencies, nearly linear behavior is observed; in the log-log plot of ρ n versus f n (see Supplemental Material [42]), we see a strong sign that ρ n ∝ f n , with the empirical robustness nearly parallel to the random expectation. As with the RNA folding PrGP maps, we suspect two causes which both contribute to this behavior: (1) as σ 2 h increases, there is a higher chance of changing the ground state, which increases phenotype entropy, and (2) a larger number of spin configurations appear as ground states, but with low frequency, drawing away probability mass from the more frequent phenotypes.
Quantum Circuit Maps.-Although methods to evolve quantum circuits have been suggested [45], to our knowledge this work is the first to analyze the structural properties of quantum circuit GP maps. We generate random quantum circuits (see Supplemental Material for algorithm) with 7 qubits and 4 layers of gates. Circuits are randomly seeded with CNOT gates which cannot participate in the genotype, and the remaining spaces are filled with single-qubit gates drawn from the alphabet {Z, X, Y, H, S, S † , T, T † }. We choose = 4 of these gates to be variable gates which comprise the genotype. The input to the circuit is the prepared state |00 . . . 0 ≡ |0 ⊗· · ·⊗|0 , and the exact probability of classically measuring the basis state |n = |qi ∈{|0 ,|1 } |q i is p n (g) = | n| U (g) |00 . . . 0 | 2 , where |q i is the i-th qubit, and U (g) is the total circuit operation. We realize these quantum circuits on the ibm lagos v1.2.0 quantum computer [42], one of the 7-qubit IBM Quantum Falcon r5.11H processors. Experimental phenotype probability vectors are constructed from tallying classical measurements from 1000 shots for each genotype. The circuits from our experimental trials are depicted in the Supplemental Material [42].
In Figure 2(e-f), we plot robustness versus frequency for each circuit output state, using both exact and experimental phenotype probability vectors for robustness calculations (see Supplemental Material [46] for Pearson and Spearman correlations and for data from additional validation trials). For the exact probabilities, the results in panel (f) strongly support the enhanced ρ n ∝ log f n scaling (Pearson r = 0.998). The spread of phenotypes in the frequency domain is due to superposition and/or entanglement; moreover, we see that many of the phentoypes are degenerate with identical frequency and robustness. This degeneracy is broken in our experimental measurements, which also exhibit measurement noise. Since we have finite shots, the degeneracies for the phenotypes observed in the exact case end up broken. The frequency and robustness of these logarithmically scaling phenotypes is suppressed relative to the exact case as probability density is drawn towards additional phenotypes, which are observed experimentally and which were not observed in the exact case. These appear due to measurement noise/decoherence effects in the physical system. The rightward shift of the phenotype entropy S(g) (see Supplemental Material [42]) further illustrates this effect.
Of the three systems investigated here, the quantum circuit PrGP map results in panel (f) are perhaps most illustrative of our suggested biphasic robustness scaling. The low frequency phenotypes which are introduced due to measurement noise in the experimental trials lie much closer to the random null expectation than the higher frequency phenotypes observed in the exact calculations, which rather scale with enhanced robustness similar to what is seen in standard DGP maps.
Discussion.-Compared to existing DGP maps, our introduction of PrGP maps not only allows for the inclusion of realistic, physical sources of disorder like thermal fluctuation and external variables, but it also permits the analysis of new systems like quantum circuits with inherent uncertainty built into the genotype-phenotype mapping and from measurement disorder. We emphasize the broad applicability of this framework to a vast array of systems across biology, physics, and computer science, and other disciplines for the analysis of robustness and stability. The proposed biphasic robustness scaling suggests that robustness of high frequency phenotypes in the DGP limit is suppressed in the PrGP formulation due to phenotype entropy increases and due to the discovery of new low frequency phenotypes. Moreover, low frequency phenotypes, which lie closer to the random null expectation, either appear randomly throughout genotype space (like in the DGP random null model), or they appear somewhat uniformly throughout a large portion of genotype space, but remain at low frequency (like in our new PrGP random null model). This scaling is observed in all three studied systems, despite being disparate, hinting at its universality. How this robustness trend affects navigability of (probabilistic) fitness landscapes is an important direction for further investigation. We also suggest that the mapping of genotypes to probability vectors instead of discrete phenotypes may facilitate the taking of gradients of, for instance, a negative loss-likelihood loss function in the process of learning PrGP or even DGP maps using statistical learning methods.
Acknowledgements.-We acknowledge the use of IBM Quantum services and the MIT Engaging Cluster for this work.
This work was supported by awards T32GM007753 and T32GM144273 from the National Institute of General Medical Sciences. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute of General Medical Sciences, the National Institutes of Health, IBM, or the IBM Quantum Team. The authors declare no known conflict of interest. * The authors contributed equally to this work. In the main text, we presented robustness versus frequency plots for RNA folding, spin glass ground state, and quantum circuit PrGP maps. For the spin glass ground state and quantum circuit PrGP maps, data for a single representative realization were presented in the main text. In Figure S1, we plot the distribution of phenotype entropy S(g) across all genotypes g for each of these PrGP maps. For RNA folding and spin glasses, we observe that the entropy distributions shift rightward as the disorder parameter increases. For RNA, this corresponds to increasing temperature and for spin glasses, this corresponds to increasing external field variance σ 2 h . For the quantum circuits, we plot both exact and experimental results (from the 7-qubit IBM quantum computer); the experimental entropy distribution is shifted rightward relative to the exact result, due to measurement noise as well as a finite number of experimental trials. In the main text, we presented robustness versus frequency and robustness versus log 10 (frequency) plots for RNA folding PrGP and DGP maps at three temperatures. For clarity, we have included robustness versus frequency, robustness versus log 10 (frequency), and log 10 (robustness) versus log 10 (frequency) plots separately for PrGP and DGP maps in Figure S2. First, we see that the DGP map results reproduce the expected ρ n ∝ log f n relationship for most phenotypes, with significant elevation above the random null model expectation. We also note that there is little temperature dependence in DGP robustness calculations, which suggests the effect of temperature does little to alter the exact ground state phenotype. In contrast, our PrGP map results showcase a different robustness behavior in which as simulation temperature increases, there is a gradual but clear suppression of the robustness versus frequency relationship; see main text for discussion of these features. In the PrGP map results we also note a biphasic behavior in which for high frequency phenotypes, the PrGP map robustness, similar to the DGP map robustness, is substantially elevated above the random null expectation and for lower frequencies, the robustness behaves more like the random model.
In Table S1, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for each map (PrGP, DGP), temperature (20°C, 37°C, 70°C), and axis transformation presented in Figure 2(a-b) and Figure S2. The primary feature we point out is the relative decrease of the PrGP Pearson r coefficients in robustness versus log 10 (frequency) plots as compared to the DGP plots; this suggests a deviation from the empirical ρ n ∝ log f n trend observed in DGP studies.
In the GP map literature, phenotype bias, the finding that phenotype frequencies can vary over many orders of magnitude with a small number of phenotypes being the targets of a large number of genotypes, has been shown for many systems [1][2][3][4]. In Figure S3, we present plots of log 10 (frequency) versus normalized rank and log 10 (frequency) versus log 10 (normalized rank) for each temperature and map pairing which show phenotype bias for this RNA folding system. Notably, the log 10 (frequency) versus log 10 (normalized rank) plot suggests a deviation from Zipf's law. Figure S4 presents transition probabilities φ mn for the most frequently occurring phenotype n to the other phenotypes m due to a single nucleotide mutation for both PrGP and DGP maps at three different temperatures. For each respective map, a plot including and excluding the most robust transition (i.e. from phenotype n → n) is shown for added clarity. This figure demonstrates that the off-diagonal transition probabilities for PrGP maps maintained an approximate relationship φ mn ∝ f m for m = n in concordance with DGP maps, and in concordance with the random null expectation for PrGP maps (see main text). A proportionality constant not equal to 1 for φ mn ∝ f m with m = n is likely due to transition probability mass that is acquired by the diagonal element φ nn . It is also apparent that the most robust transition is much more likely than the transition to any other phenotype, in support of our claim that PrGP maps, like DGP maps, exhibit enhanced robustness.   Here, we present results of a validation trial for RNA folding PrGP maps for sequences of length = 12 utilizing the full alphabet of size k = 4, {A, C, G, U}. In Figure S5, we present robustness versus frequency, robustness versus log 10 (frequency), and log 10 (robustness) versus log 10 (frequency) plots for RNA folding for PrGP and DGP maps. As with the reduced alphabet case, we see both PrGP and DGP map results show significant elevation above the random null model expectation, with PrGP map results demonstrating a gradual but clear suppression of the robustness versus frequency relationship compared to DGP map results. The expected ρ n ∝ log f n relationship for phenotypes in the DGP map results as well as the biphasic behavior of the PrGP map results is present but less clear in this case, likely due to a small size effect from the limited number of phenotypes present in this complete alphabet (k = 4, = 12) system compared to the reduced alphabet system (K = 2), which contains sequences of longer length ( = 20). Also in Figure S5, we plot the distribution of phenotype entropy S(g) across all genotypes g; most phenotype entropies are zero due to their being deterministic because for the RNA folding, k = 4, = 12 system most genotypes do not fold.
In Table S2, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for each map (PrGP, DGP) and axis transformation presented in Figure S5. In Figure S6, we present plots of log 10 (frequency) versus normalized rank and log 10 (frequency) versus log 10 (normalized rank) for each temperature and map pairing which show phenotype bias for this RNA folding system. Notably, the log 10 (frequency) versus log 10 (normalized rank) plot suggests a deviation from Zipf's law. Figure S7 presents transition probabilities φ mn for the most frequently occurring phenotype n to the other phenotypes m due to a single nucleotide mutation for both PrGP and DGP maps. For each respective map, a plot including and excluding the most robust transition is shown for added clarity. This figure demonstrates that the off-diagonal transition probabilities for PrGP maps maintained an approximate relationship φ mn ∝ f m for m = n in concordance with DGP maps, and in concordance with the random null expectation for PrGP maps (see main text). A proportionality constant not equal to 1 for φ mn ∝ f m with m = n is likely due to transition probability mass that is acquired by the diagonal element φ nn . It is also apparent that the most robust transition is much more likely than the transition to any other phenotype, in support of our claim that PrGP maps, like DGP maps, exhibit enhanced robustness.

IV. EXTENDED DATA FOR MAIN TEXT SPIN GLASS PrGP MAP
In the main text, we compared a spin glass DGP map with a fixed random external field {h 0,i } with our spin glass PrGP map, which introduces a Gaussian distribution to the external field whose means are fixed at {h 0,i } and whose variance σ 2 h is varied as an independent variable. Figure S8 shows the topology of the graph G(V, E) (with |V | = 9, |E| = 15) that corresponds to the spin glass PrGP map data presented in the main text.
Main text Figure 2(c, d) presents robustness versus frequency and robustness versus log 10 (frequency) data; Figure S9 additionally plots log 10 (robustness) versus log 10 (frequency) for these same data. These three plots collectively demonstrate that, as with RNA folding GP maps, as the disorder parameter increases the uncertainty in the genotypephenotype pairing, the robustness versus frequency relationship in PrGP maps becomes suppressed relative to the DGP map limit. These spin glass results are highly suggestive of a biphasic robustness relationship where at high frequencies, ρ n is substantially enhanced above the random null expectation and behavior close to the deterministic limit is observed. However, as is clear from Figure S9, nearly linear behavior is observed for the smallest frequencies with the empirical robustness nearly parallel to the random expectation, suggesting ρ n ∝ f n . See main text for discussion of these features.
In Table S3, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for each map (PrGP, DGP), external field variance (σ 2 h = 0.001, σ 2 h = 0.01, σ 2 h = 0.1), and axis transformation presented in Figure 2(c, d) and Figure S9. The primary feature we point out is the relative decrease of the PrGP Pearson r coefficients in robustness versus log 10 (frequency) plots as compared to the DGP (deterministic) plot; this suggests a deviation from the empirical ρ n ∝ log f n trend observed in the spin glass DGP study [5].
In Figure S10, we present plots of log 10 (frequency) versus normalized rank and log 10 (frequency) versus log 10 (normalized rank) for each external field variance and the deterministic case. Notably, the log 10 (frequency) versus log 10 (normalized rank) plot suggests a deviation from Zipf's law. Figure S11 presents transition probabilities φ mn for the most frequently occurring ground state n to the other ground states m due to a single bond perturbation. For each setting of external random field variance, a plot including and excluding the most robust transition is shown for added clarity. This figure demonstrates that the off-diagonal transition probabilities for PrGP maps maintained an approximate relationship φ mn ∝ f m for m = n in concordance with DGP maps, and in concordance with the random null expectation for PrGP maps (see main text). A proportionality constant not equal to 1 for φ mn ∝ f m with m = n is likely due to transition probability mass that is acquired by the diagonal element φ nn . It is also apparent that the most robust transition is much more likely than the transition to any other phenotype, in support of our claim that PrGP maps, like DGP maps, exhibit enhanced robustness.     TABLE S3. Pearson and Spearman correlation coefficients for all robustness versus frequency plots for the spin glass PrGP map with |V | = 9 and |E| = 15 whose data are shown in the main text and here in the Supplemental Material.
FIG. S10. Plots of (left) log10(frequency) versus normalized rank and (right) log10(frequency) versus log10(normalized frequency) for spin glass ground states for PrGP maps at three external field variances and DGP maps the deterministic case. When computing ranks, ties were broken arbitrarily.

V. VALIDATION TRIAL FOR SPIN GLASS PrGP MAP
We provide a second spin glass PrGP map trial here in the Supplemental Material to illustrate that the spin glass trends described above and in the main text hold across multiple random graph instances. We generate a new G(V, E), once again with |V | = 9, |E| = 15 with topology shown in Figure S12. Figure S13 presents robustness versus frequency, robustness versus log 10 (frequency), and log 10 (robustness) versus log 10 (frequency) for spin glass PrGP maps at three different external field variances and for the deterministic case for DGP maps. The results from this validation trial exhibit the same behavior as observed in the trial presented in the main text. In particular, we see that as the disorder parameter increases the uncertainty in the genotype-phenotype pairing, the robustness versus frequency relationship in PrGP maps becomes suppressed relative to the DGP map limit. Again, these spin glass results are highly suggestive of a biphasic robustness relationship where at high frequencies, ρ n is substantially enhanced above the random null expectation and behavior close to the deterministic limit is observed. However, as is clear from Figure S9, nearly linear behavior is observed for the smallest frequencies with the empirical robustness nearly parallel to the random expectation, signaling ρ n ∝ f n . See main text for discussion of these features. Additionally, Figure S13 plots the distribution of phenotype entropy S(g) across all genotypes g for PrGP maps at each external field variance experimental value. As is the case in Figure S1, we observe that the entropy distributions shift rightward as the disorder parameter increases.
In Table S4, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for each map (PrGP, DGP), external field variance (σ 2 h = 0.001, σ 2 h = 0.00, σ 2 h = 0.1), and axis transformation presented in Figure S13. The primary feature we point out is the relative decrease of the PrGP Pearson r coefficients in robustness versus log 10 (frequency) plots as compared to the DGP (deterministic) plot; this suggests a deviation from the empirical ρ n ∝ log f n trend observed in the spin glass DGP study [5]. FIG. S13. Plots of (leftmost) robustness versus frequency, (middle left) robustness versus log10(frequency), and (middle right) log10(robustness) versus log10(frequency) for spin glass ground states for PrGP maps at three different external field variances and for the deterministic case for DGP maps. Additionally, the (rightmost) density versus phenotype entropy for the spin glass ground states at three difference external field variances is plotted. The dashed line is the random null expectation for both PrGP and DGP maps given by φmn = fm for all m and n.  TABLE S4. Pearson and Spearman correlation coefficients for all robustness versus frequency plots for the spin glass PrGP map validation trial with |V | = 9 and |E| = 15 whose data are shown above in this section.

VI. QUANTUM CIRCUIT GENERATION ALGORITHM
In this study, we generated quantum circuits with 7 qubits and 4 layers. We take the genotype of the quantum circuit PrGP map to be a subset of single qubit gates (which are varied to reflect each genotype). We first start by seeding the circuit randomly with CNOT gates which cannot participate in the genotype gate list. Only certain pairs of qubits which are physically connected in the 7-qubit ibm lagos v1.2.0 quantum computer can participate in the same CNOT gate. The remaining open places are seeded with single qubit gates, and we choose = 4 of these gates to be the variable gates for the genotype. The alphabet chosen is of size k = 8: {Z, X, Y, H, S, S † , T, T † }. Circuit diagrams used in our experimental trials are shown in the subsequent sections.

VII. EXTENDED DATA FOR MAIN TEXT QUANTUM CIRCUIT PrGP MAP
To our knowledge, this work is the first to analyze the structural properties of quantum circuit GP maps. We generate random quantum circuits as described in the main text and in the previous section with 7 qubits and 4 layers of gates. Figure S14 shows a schematic representation of the random quantum circuit generated for the quantum circuit PrGP map data presented in main text Figure 2(e, f) and in this Supplemental Material section.
Main text Figure 2(e, f) presents robustness versus frequency and robustness versus log 10 (frequency) data using both exact and experimental phenotype probability vectors for robustness calculations; Figure S15 additionally plots log 10 (robustness) versus log 10 (frequency) for these same data. Collectively, we see that for the exact probabilities, the results strongly support the enhanced ρ n ∝ log f n scaling. The spread of phenotypes observed in the frequency domain is due to superposition and/or entanglement and many of the phentoypes are degenerate with identical frequency and robustness. This degeneracy is broken in our experimental measurements, which also exhibit measurement noise. Moreover, the frequency and robustness of these logarithmically scaling phenotypes is suppressed relative to the exact case as probability density is drawn towards additional phenotypes which are observed experimentally which were not observed in the exact case. These quantum circuit PrGP map results are perhaps most illustrative of our suggested biphasic robustness scaling. The low frequency phenotypes which are introduced due to measurement noise in the experimental trials lie much closer to the random null expectation than the higher frequency phenotypes observed in the exact calculations, which rather scale with enhanced robustness similar to what is seen in standard DGP maps.
In Table S5, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for both exact and experimental quantum circuit PrGP results for each axis transformation presented in main text Figure 2(e, f) and Figure S15. The primary features we point out are the high Perason correlation r = 0.998 of the robustness versus log 10 (frequency) relationship for the exact phenotype probability vectors, and the relative decrease of the experimental Pearson r coefficients in robustness versus log 10 (frequency) plot as compared to the exact plot. This suggests that the exact relationship exhibits behavior similar to the empirical ρ n ∝ log f n trend observed in DGP studies, and that the experimental trials introduce measurement noise which induces a deviation from the exact results.
In Figure S16, we present plots of log 10 (frequency) versus normalized rank and of log 10 (frequency) versus log 10 (normalized rank) for experimental and exact quantum circuit PrGP map results. Notably, the plot showing log 10 (frequency) versus log 10 (normalized rank) suggests a deviation from Zipf's law. Figure S17 presents transition probabilities φ mn for the most frequently occurring circuit output state n to the other circuit output states m due to a single variable gate perturbation. For both experimental and exact phenotype probability vectors, a plot including and excluding the most robust transition is shown for added clarity. This figure demonstrates that the off-diagonal transition probabilities for quantum circuit PrGP maps are positively correlated with the frequency f m , though there appears to be some additional nonrandom relationship which is not predicted from standard DGP or PrGP theory. It is also apparent that the most robust transition is much more likely than the transition to any other phenotype, in support of our claim that PrGP maps, like DGP maps, exhibit enhanced robustness.  TABLE S5. Pearson and Spearman correlation coefficients for all robustness versus frequency plots quantum circuit PrGP map whose robustness data was presented in the main text and in the above log-log plot. This includes both exact results as well as experimental results for realization/Trial 1, whose circuit is also printed earlier in this section.

VIII. VALIDATION TRIALS FOR QUANTUM CIRCUIT PrGP MAP
To validate the quantum circuit PrGP map results presented in the main text and Supplemental Material, six additional trials were conducted. A schematic of the random quantum circuit generated for the first of these validation trials is shown in Figure S18. Figure S19 presents robustness versus frequency, robustness versus log 10 (frequency), and log 10 (robustness) versus log 10 (frequency) for this quantum circuit PrGP map validation trial. As with the first quantum circuit PrGP map trial, these data strongly support the enhanced ρ n ∝ log f n scaling. Again, we see the spread of phenotypes observed in the frequency domain due to superposition and/or entanglement and that many of the phentoypes are degenerate with identical frequency and robustness. This degeneracy is broken in our experimental measurements, which exhibit measurement noise. Once again, the frequency and robustness of these logarithmically scaling phenotypes is suppressed relative to the exact case as probability density is drawn towards additional phenotypes which are observed experimentally which were not observed in the exact case. These results illustrate our suggested biphasic robustness scaling in which the low frequency phenotypes, which are introduced due to measurement noise in the experimental trials, lie much closer to the random null expectation than the higher frequency phenotypes observed in the exact calculations, which rather scale with enhanced robustness similar to what is seen in standard DGP maps. Figure S19 also presents a plot of the distribution of phenotype entropy S(g) across all genotypes g for exact and experimental quantum circuit PrGP maps. Notably, the experimental entropy distribution is shifted rightward relative to the exact result due to measurement noise as well as a finite number of experimental trials.
In Table S6, we include the Pearson correlation coefficient r and Spearman rank correlation coefficient ρ for both exact and experimental quantum circuit PrGP results for each axis transformation presented in Figure S19. The primary features we point out are the high Perason correlation r = 0.950 of the robustness versus log 10 (frequency) relationship for the exact phenotype probability vectors, and the relative decrease of the experimental Pearson r coefficients in robustness versus log 10 (frequency) plot as compared to the exact plot. This suggests that the exact relationship exhibits behavior similar to the empirical ρ n ∝ log f n trend observed in DGP studies, and that the experimental trials introduce measurement noise which induces a deviation from the exact results. Figure S20 presents robustness versus frequency and robustness versus log 10 (frequency) plots as well as schematics of the corresponding random quantum circuits for validation trials 3-7. In each trial, the suggested biphasic robustness scaling is clear. Additionally, these trials support the enhanced ρ n ∝ log f n scaling.

Trial 2 (Validation)
FIG. S18. Random circuit generated for quantum circuit trial 2, whose robustness and entropy data are plotted below as a validation trial. The genotype is the set of variable gates g = (G0, G1, G2, G3), so the length of the input sequence is = 4 drawn from an alphabet of k = 8 single qubit gates: {Z, X, Y, H, S, S † , T, T † }.
FIG. S19. Plot of (leftmost) robustness versus frequency, (left middle) robustness versus log10(frequency), and (middle right) log10(robustness) versus log10(frequency) for quantum circuit trial 2 for experimental and exact data. Additionally, the (rightmost) density versus phenotype entropy for quantum circuit trial 2 is plotted. The dashed line is the random null expectation for both PrGP and DGP maps given by φmn = fm for all m and n.

System
Map  TABLE S6. Pearson and Spearman correlation coefficients for all robustness versus frequency plots quantum circuit PrGP map whose robustness data is shown above as a Validation trial (i.e. Trial 2). This includes both exact results as well as experimental results for Trial 2, whose circuit is also printed earlier in this section.